10.1.2Does the series diverge?

The Divergence Test

10-12.

Summarize what it means for an infinite series to be convergent or divergent. Use the term “partial sum” in your explanation.

10-13.

What happens to the $n$ ^th term?

Consider the series: $–2 + –1 + 0 + 1 + 2 + …$ . Does this series converge or diverge?
Use the limit of the sequence of partial sums to support your answer.
What about $\frac { 1 } { 2 } + \frac { 1 } { 2 } + \frac { 1 } { 2 } + \frac { 1 } { 2 } + \ldots$ ? Does this series converge or diverge? Why or why not?
Does $- 2 - 1 \frac { 1 } { 2 } - 1 \frac { 1 } { 4 } - 1 \frac { 1 } { 8 } -$ converge or diverge? Why or why not?
Compare the infinite series in parts (a) through (c). If $n$ represents the number of terms in a series, what do you notice about the $n$ ^th term of each of these infinite series as $n$ approaches infinity?
Copy and complete the statement below to write a conjecture about how to use the $n$ ^th term of an infinite series (as $n$ approaches infinity) to determine if the series will diverge.

The Divergence Test
(a.k.a. The $\mathbf{n^{th}}$ Term Test)

For some series $S = \displaystyle \sum _ { k = 0 } ^ { \infty } a_k$ , if ________________, the $S$ diverges.

10-14.

Connie is wondering whether the inverse of the Divergence Test will guarantee that a series converges. In other words, she wonders if a series has the opposite characteristics as the series in problem 10-13, will it definitely converge?

Write the inverse of Connie’s conjecture in if…, then… form.
Conjecture: If __________________, then $S$ converges.
Remember that this is just a conjecture!
Try to disprove Connie’s conjecture. In other words, can you think of an infinite series that is a counterexample to her if…, then… statement.
It may surprise you to learn that there are many counterexamples to Connie’s convergence conjecture. Lessons 10.1.3 through 10.1.8 will introduce ways to recognize these counterexamples. For now, it is important to remember that the Divergence Test guarantees that a series will diverge. However, failing the Divergence Test does not guarantee that the series will converge.

With your team, think of three examples (from your life) of if…, then…. statements that are true, but the inverse is not necessarily true.

10-15.

Which series from problem 10-2 diverge based on the Divergence Test?

10-16.

Rewrite the series $S = \frac { 1 } { 2 \cdot 3 } + \frac { 1 } { 3 \cdot 4 } + \frac { 1 } { 4 \cdot 5 } + \ldots$ three different ways using sigma notation by completing the following expressions: Homework Help ✎

$S = \displaystyle\sum _ { n = 1 } ^ { \infty } (\ \ \ )$

$S = \displaystyle \sum _ { n = 0 } ^ { \infty } (\ \ )$

$S = \displaystyle \sum _ { n = 5 } ^ { \infty } (\ \ \ )$

10-17.

For each of the series below, decide if there is a finite sum. If there is a finite sum, predict the sum. If there is not a finite sum, explain why. Homework Help ✎

$5 + 10 + 15 + 20 + …$

$0.1 + 0.01 + 0.001 + 0.0001 + …$

$4+\frac{4}{3}+\frac{4}{9}+\frac{4}{27}+...$

$2 − 2 + 2 − 2 + …$

10-18.

Jenny made the following table while graphing a polar function. Write an equation for the function and describe its shape. Homework Help ✎

$θ$

$0$

$\frac{π}{6}$

$\frac{π}{4}$

$\frac{π}{3}$

$\frac{π}{2}$

$\frac{2π}{3}$

$\frac{3π}{4}$

$\frac{5π}{6}$

$π$

$\frac{7π}{6}$

$\frac{5π}{4}$

$\frac{4π}{3}$

$\frac{3π}{2}$

$\frac{5π}{3}$

$\frac{7π}{4}$

$\frac{11π}{6}$

$2π$

$\mathbf{r}$

$0$

$\frac{π}{12}$

$\frac{π}{8}$

$\frac{π}{6}$

$\frac{π}{4}$

$\frac{π}{3}$

$\frac{3π}{8}$

$\frac{5π}{12}$

$\frac{π}{2}$

$\frac{7π}{12}$

$\frac{5π}{8}$

$\frac{2π}{3}$

$\frac{3π}{4}$

$\frac{5π}{6}$

$\frac{7π}{8}$

$\frac{11π}{12}$

$π$

10-19.

Sketch a slope field for $\frac { d y } { d x } = y ( 10 - y )$ for $0 ≤ y ≤ 10$ . Sketch a particular solution that goes through $(0, 0)$ . 10-19 HW eTool (Desmos). Homework Help ✎

10-20.

Examine the integrals below. Consider the multiple tools available for integrating and use the best strategy for each part. Evaluate each integral, and briefly describe your method. Homework Help ✎

$\int _ { 1 } ^ { 4 } \sqrt { 1 + \frac { 9 } { 4 } x } d x$

$\int 2 x \operatorname { csc } ( x ^ { 2 } ) \operatorname { cot } ( x ^ { 2 } ) d x$

$\int \frac { e ^ { 1 / x } } { 2 x ^ { 2 } } d x$

$\int _ { - 10 } ^ { 10 } \operatorname { cos } ( 0 ) d x$

10-21.

Write the equation of the curve whose derivative at every point $(x, y)$ is $\frac { 3 x ^ { 2 } } { 2 y }$ and passes through the point $(2, 1)$ . Homework Help ✎

10-22.

Does the series $400 - 200 + 100 - 50 + …$ converge? If so, to what value? If not, why not? Homework Help ✎

10-23.

Use the trigonometric identity $\cos(2x) = 2\cos^2(x) - 1$ to rewrite each of the expressions below. Homework Help ✎

$\cos^2(x)$

$\sin^2(x)$

Calculus Third Edition